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Independence and duality. (English) Zbl 0782.08001

Let \(J: P(M)\mapsto P(M)\) be a closure operator on \(M\). A subset \(X\subseteq M\) is called \(J\)-independent if for all \(Y,Z\subseteq X\): \(J(Y)= J(Z)\Rightarrow Y= Z\). In this paper, the author formulates some properties of \(J\)-independence and generalizes them to a mapping (even to an arbitrary binary relation). The relational (functional) properties reveal the duality character of independence and the generating-process.

MSC:

08A02 Relational systems, laws of composition
06A15 Galois correspondences, closure operators (in relation to ordered sets)
03E20 Other classical set theory (including functions, relations, and set algebra)
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